Vagueness, the Sorites Paradox, and Precisifications

In the (shorter) first half of Chap. 3 , the Boolean many-valued solution to the Sorites Paradox is given. The (much longer) second half investigates the relation between the Boolean many-valued approach and another, popular, approach to vagueness, the precisificational approach. It is shown that the Boolean value of a sentence can be identified with the set of precisifications in which the sentence is true. A notion of logical consequence, called the reformulated bottomline preservation notion, can be introduced directly to precisificational spaces; alternatively, the S4 determinacy (or ‘determinately’) operator can be introduced into the spaces with a bivalent semantics. These three approaches, i.e., the Boolean many-valued approach, the precisificational approach with the reformulated bottomline preservation notion of logical consequence, and the modal, S4 determinacy operator approach, are shown equivalent. Supervaluationism, of both global and local variety, is dismissed as employing incorrect notions of logical consequence. Another modal operator, the S5 determinacy operator, is introduced.